Generalized Coordinates and Constraints, Physics tutorial

Introduction:

The rigid body is stated as the system of n particles for which all interparticle distances are constrained to fixed constants, |ri - rj| = cij and interparticle potentials are functions only of these interparticle distances. As these distances don't differ, neither does internal potential energy. These interparticle forces can't do work, and internal potential energy may be ignored.

Rigid body is the example of the constrained system, in which general 3n degrees of freedom are limited by some forces of constraint that place conditions on coordinates ri may be in conjunction with their momenta.

Degrees of Freedom:

Number of degrees of freedom is stated as number of independent coordinates which is required to recognize uniquely configuration of system.

Assume we have the system of N particles each moving in 3-space and interacting through arbitrary (finite) forces. The number n=3N is known as number of degree of freedom. There is freedom, of course, in how we state degrees of freedom; like:

  • Choice of origin
  • Coordinate system: Cartesian, cylindrical, spherical, etc.
  • Center-of-mass vs. individual particles :{rk} or {R, Sk = rk = rk - R}

But, number of degrees of freedom is always same; like, in center-of-mass system, constraint Σk Sk = 0 applies, ensuring that{rk} and R, Sk have same number of degrees of freedom. Motion of such a system is entirely specified by knowing dependence of available degrees of freedom on time.

Constraints:

Constraints mean the limitation on degree of freedom of motion of the system of particles in form of a condition. Constraints may decrease number of degrees of freedom; like, particle moving on the table, rigid body, etc. It is separated in Holonomic and Non-Holonomic constraint.

Holonomic Constraints:

Holonomic constraints are those which can be expressed in form f(r1, r2,.....t) = 0. For instance, limiting a point particle to move on surface of the table is holonomic constraint z - zo = 0, where z0 is the constant. The rigid body satisfies holonomic set of constraints |ri - rj| - cij = 0, where cij is the set of constants satisfying cij = cji>0 for all particle pairs i, j. The system is known as holonomic if, in the certain sense, one can recover global information from local information, so meaning entire-law is fairly suitable.

Holonomic constraints may be separated in rheonomic i.e. running law) and scleronomic i.e. rigid law depending on whether time seems explicitly in constraints:

Rheonomic: f({rk}, t) = 0,

Scleronomic: f({rk}) = 0

At the technical level, difference is whether ∂f/∂t = 0 or not: presence of the partial derivative affects some of the relations.

Non-holonomic Constraints:

Rolling of the ball on the table is non-holonomic, as one rolling along different paths to same point can put it in different orientations.

Non-holonomic constraints are, obviously, constraints which are not holonomic. There are three types of non-holonomic constraints:

i) Non-integrable or history-dependent constraints. These are constraints which are not completely stated until full solution of equations of motion is known. Equivalently, they are certain kinds of constraints involving velocities. Classic case of this kind is the vertical disk rolling on the horizontal plane. If x and y define position of point of contact between disk and plane, Φ states angle of rotation of disk about its axis, and Θ states angle between rotation axis of the disk and the x-axis, then one can find constraints

x = -r Φ cosΘ,

y = -r Φ sinΘ.

The differential version of the constraints is

dx = -r dΦ cosΘ,

dy = -r dΦ sinΘ.

These differential equations are not integrable; one cannot generate from the relations two equations f1(x, Θ, Φ ) =0 and f2(y, Θ, Φ) = 0. The reason is that, if one assumes the functions f1 and f2 exist, the above differential equations imply that their second derivatives would have to satisfy

2f/∂Θ∂Φ ≠ ∂2f/∂Φ∂Θ i.e. for f = f1 or f2

This is very unpleasant mathematical condition. Explicitly, assume f1 existed. Then we would be able to write f1(x, Θ, Φ ) =0

Let us get differential version of constraint by differentiation:

(∂f1/∂x)dx + (∂f1/∂Θ)dΘ + (∂f1/∂Φ)dΦ = 0

This differential constraint must match original differential constraint dx = -rdΦcosΘ. Identifying coefficients of differential yields

∂f1/∂x = 1 ∂f1/∂Θ = 0 ∂f1/∂Φ= rcosΦ

Taking mixed second partial derivatives provides

2f1/∂Φ∂Θ = 0, ∂2f1/∂Θ∂Φ = -rsinΦ

This, clearly, do not match. Such constraints are also known as non-integrable as one can't integrate differential equation to find the constraint on coordinates. The differential relation like one above is a local one; if differential relation is integrable, you can get constraint at all points in space that is you can find entire law. Clearly, non-integrability is also related to fact that constraint is velocity-dependent: a velocity-dependent constraint is the local constraint, and it may not always be possible to find out global constraint from it.

ii. Inequality constraints; like particles needed to stay inside the box, particle sitting on the sphere but allowed to roll off,

iii. Problems involving frictional forces.

Generalized coordinates:

In general, if one has j independent constraint equations for a system of M particles with 3M degrees of freedom, then the true number of degrees of freedom is 3M - j. There is dynamical behavior of system in only these remaining degrees of freedom.

For holonomic constraints, constraint equations make sure that it will always be possible to identify the new set of 3M - j generalized coordinates {qk} which completely specify motion of system subject to constraints and that are independent of each other. Independence arises as number of degrees of freedom should be conserved. Constraint equations yield (possibly implicit) functions

r 1 = r1(q1, q2,....q3M-j, t)

That transform between generalized coordinates and original coordinates. It may not always be possible to write the functions analytically. Few of the coordinates may be same as original coordinates, some may not; it just depends on structure of constraints. Generalized coordinates are more than just the notational convenience. By including constraints in definition of generalized coordinates, we get two significant simplifications:

1) Constraint forces are eliminated from problem; and

2) Generalized coordinates are completely independent of each other.

Dot Cancellation:

For holonomic constraints, there is very significant statement that we will make much use of later

∂r/∂qk = ∂dr/∂dqk

This is the differential relationship between given degree of freedom and generalized coordinate is same as the differential relationship between corresponding velocities. This statement depends on having holonomic constraints.

Just as velocity corresponding to the coordinate rj is rj = d/dtrj, it is possible to state the generalized velocity qk as qk = d/dtqk. Note that in all cases, velocities are stated as total time derivatives of particular coordinate. Remember that if you have function g = g({qk}, t) then total time derivative d/dt g is evaluated by chain rule:

d/dt g({qk}, t) = Σ∂g/∂qk dqk/dt + ∂g/∂t

It is very significant to realize that, until the specific solution to equation of motion is found, a generalized coordinate and its corresponding generalized velocity are independent variables. This can be seen by just remembering that two initial conditions that are qk(t = 0) and qk(t = 0) are needed to state a solution of Newton's second law, as it is a second-order differential equation. Higher-order derivatives are not independent variables as Newton's second law relates higher-order derivatives to {qk} and {qk}. Independence of {qk} and {qk} is the reflection of structure of Newton's second law, not just mathematical theorem. Unless otherwise indicated, from here on we will suppose all constraints are holonomic.

Now, take partial derivative with respect to 1 q1; this selects out term in sum with

k = l, and drops t term:

∂dr/∂dq1 = Σk∂ri/∂qk δkl = ∂ri/∂q1

So dot cancellation relation is proven.

Virtual displacement, virtual work and generalized forces:

Virtual Displacement:

We state virtual displacement {δri } as the infinitesimal displacement of system coordinates {ri} that satisfies given criteria.

1. Displacement satisfies constraint equations, but may make use of any remaining unconstrained degrees of freedom.

2. Time is held fixed during displacement.

3. Generalized velocities {qk } are held fixed during displacement.

The virtual displacement can be signified in terms of position coordinates or generalized coordinates. Benefit of generalized coordinates, of course, is that they automatically respect constraints. The arbitrary set of displacements δqk dq can qualify as the virtual displacement if conditions (2) and (3) are additionally applied, but the arbitrary set of displacements{δri} may or may not qualify as the virtual displacement depending on whether displacements obey constraints. All three conditions will become clearer in examples.

Explicitly, relationship between infinitesimal displacements of generalized coordinates and virtual displacements of position coordinates is

δri = Σk(∂ri/∂qk)δqk.............................Eq.1

This expression has content: there are fewer {qk} than { ri}so the fact which can be stated only in terms of the {δri} reflects fact that virtual displacement respects constraints. One can put in any values of δqk and obtain the virtual displacement, but not every possible set of {δri} can be written in above way.

Virtual Work:

Using virtual displacement, we state virtual work as work which would be done on system by forces applying on system as system goes through virtual displacement {δri}

δW ≡ ΣijFijri..........................................................Eq.2

Where Fij is jth force applying on coordinate of ith particle ri

Mathematically, assumption lets us drop part of the virtual work sum containing constraint forces, leaving

δW = ΣijFij(nc).δri..........................................................Eq.3

Generalized Force:

To state generalized forces, we combine Equation 1, relationship between virtual displacements of position coordinates and generalized coordinates, with Equation 3, relationship between virtual work and non-constraint forces:

fk ≡ Σij Fij(nc).∂ ri/∂qk = δW/δqk

It is generalized force along the kth generalized coordinate. Last expression, fk = δW/δqk says that force is simple the ratio of work done to displacement when a virtual displacement of only the kth generalized coordinate is performed; it is of course possible to displace only kth generalized coordinate as generalized coordinates are mutually independent. It is significant to remember that generalized force is found by summing only over non-constraint forces: constraint forces have already been taken in account in expressing generalized coordinates. Infinitesimally, a force F causing the displacement δr does work δW = F. δr. Generalized force is exact analogue: if work δW is done when ensemble of forces act to generate the generalized coordinate displacement δqk, then generalized force fk doing that work is fk = δW/δqk. But generalized force is the simplification as it is only made up of non-constraint forces.

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